Estudio de las componentes de Fatou de funciones racionales sobre la esfera de Riemann

Abstract

This master thesis is focused on the study of Fatou set of rational functions on the Riemann sphere. In these notes, we classify the components of the Fatou set and we prove the No Wandering Domains Theorem. We also conclude this work with an example of an entire function whose Fatou set has wandering domains. This manuscript is a continuation of a previous degree thesis I presented in 2022 which was devoted to the study of Julia sets of rational functions. Therefore, in the first chapter, we include a summary of the notions covered in the degree thesis that will be treated as elementary throughout the remaining of this work. The second chapter aims to provide some knowledge about Fatou set structure. In this chapter we study the Euler characteristic for regular subdomains of the Riemann sphere to establish the Riemann-Hurwitz relation for Fatou set components. We also bound the number of invariant connected components that a Fatou set of a rational function might have. In the third chapter, we classify periodic points and cycles of an analytic function which are immediately applied to prove various results concerning the local behaviour of rational functions at these points. In Chapters 4 and 5, we classify completely Fatou connected components. We shall remark one significant result from each chapter: in Chapter 4, Theorem 4.1.2, which establishes that a forward invariant component of the Fatou set is one of five possibilities and; in the fifth chapter, the No Wanderings Domains Theorem by D. Sullivan which assures that every component of the Fatou set of a rational map is eventually periodic. Finally, in the sixth chapter, we provide an example of an entire function which has wandering domains. This example was given by I. N. Baker, in 1974, before Sullivan proved his theorem.